Vector subtraction is a fundamental operation in linear algebra that allows us to find the difference between two vectors. It is performed by subtracting the corresponding components of the vectors.

This operation is useful in various applications, such as physics, engineering, and computer graphics.

In this operation we combines two vectors to create a new vector. It is performed by subtracting the corresponding components of the vectors.

## Mathematical Method

Let’s consider two vectors $\mathbf{A}$ and $\mathbf{B}$:

\[

\mathbf{A} = \begin{pmatrix} A_1 \\ A_2 \\ A_3 \end{pmatrix} \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} B_1 \\ B_2 \\ B_3 \end{pmatrix}

\]

The subtraction of $\mathbf{B}$ from $\mathbf{A}$ is denoted as $\mathbf{A} – \mathbf{B}$, and it can be calculated as:

\[

\mathbf{A} – \mathbf{B} = \begin{pmatrix} A_1 – B_1 \\ A_2 – B_2 \\ A_3 – B_3 \end{pmatrix}

\]

## Geometric Method

The geometric method for vector subtraction involves visualizing vectors as directed line segments in space and using their geometric properties to perform the subtraction.

Let’s consider two vectors $\mathbf{A}$ and $\mathbf{B}$.

To subtract vector $\mathbf{B}$ from vector $\mathbf{A}$ using the geometric method, follow these steps:

- Draw vector $\mathbf{A}$ as a directed line segment starting from the origin.
- Draw vector $\mathbf{B}$ as a directed line segment starting from the same initial point as $\mathbf{A}$.
- Extend vector $\mathbf{B}$ in the opposite direction until it intersects the line representing vector $\mathbf{A}$.
- The vector from the origin to the point of intersection is the resultant vector, which represents $\mathbf{A} – \mathbf{B}$.

## Example

Let’s consider the following vectors:

\[

\mathbf{A} = \begin{pmatrix} 2 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}

\]

To find $\mathbf{A} – \mathbf{B}$, we subtract the corresponding components:

\[

\mathbf{A} – \mathbf{B} = \begin{pmatrix} 2 – 1 \\ 4 – 3 \\ 1 – 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}

\]

Therefore, the result of $\mathbf{A} – \mathbf{B}$ is $\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$.